
where the \(\otimes \)  symbol in the second equation signals that a proper ordering of the two matrices must be taken into account. To all orders in perturbation theory the anomalous dimensions take the form [1]}, [2]}
\(\Gamma _{\phi \phi }(\mu )&= \left( \frac{1}{4}\,\gamma _\mathrm {cusp}^{(1)} + \frac{3}{4}\,\gamma _\mathrm {cusp}^{(2)} \right)\left( \ln \frac{m_{Z^{\prime }}^2}{\mu ^2} - i\pi \right) + 2\gamma ^\phi \,, \\\mathbf {\Gamma }_{Q_L Q_L}(\mu )&= \left( \frac{1}{36}\,\gamma _\mathrm {cusp}^{(1)} + \frac{3}{4}\,\gamma _\mathrm {cusp}^{(2)}+ \frac{4}{3}\,\gamma _\mathrm {cusp}^{(3)} \right)\left( \ln \frac{m_{Z^{\prime }}^2}{\mu ^2} - i\pi \right) + \lbrace  \mathbf {\gamma }^{Q_L}, \,.\, \rbrace  \,, \\\mathbf {\Gamma }_{L_L L_L}(\mu )&= \left( \frac{1}{4}\,\gamma _\mathrm {cusp}^{(1)} + \frac{3}{4}\,\gamma _\mathrm {cusp}^{(2)} \right)\left( \ln \frac{m_{Z^{\prime }}^2}{\mu ^2} - i\pi \right) + \lbrace  \mathbf {\gamma }^{L_L}, \,.\, \rbrace  \,, \\\mathbf {\Gamma }_{q_R q_R}(\mu )&= \left( e_q^2\,\gamma _\mathrm {cusp}^{(1)} + \frac{4}{3}\,\gamma _\mathrm {cusp}^{(3)} \right)\left( \ln \frac{m_{Z^{\prime }}^2}{\mu ^2} - i\pi \right) + \lbrace  \mathbf {\gamma }^{q_R}, \,.\, \rbrace  \,; \quad q=u,d ,\\\mathbf {\Gamma }_{e_R e_R}(\mu )&= \gamma _\mathrm {cusp}^{(1)} \left( \ln \frac{m_{Z^{\prime }}^2}{\mu ^2} - i\pi \right)+ \lbrace  \mathbf {\gamma }^{e_R}, \,.\, \rbrace  \,. \) 
